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# Staggered Difference-in-Differences in Corporate Research
## Methodological Challenges and a Path Forward
### Andrew C. Baker, David. F. Larcker, and Charles C.Y. Wang
### Stanford GSB and Harvard HBS
### Law and Economics Virtual Seminar <br/> December 16, 2020

---


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# .center.pull[Outline of Talk]

`\(\hspace{2cm}\)`

1. Basics of Difference-in-Differences

2. Regression Difference-in-Differences

3. Problems with Staggered Difference-in-Differences

4. Simulation Results

5. New Difference-in-Differences Methods

6. Banking Law Application

---

# .center.pull[Difference-in-Differences]

`\(\hspace{2cm}\)`

- Observational analog to an RCT with multiple time periods of observation.

`\(\hspace{2cm}\)`

- 2 units and 2 time periods.

`\(\hspace{2cm}\)`

- 1 unit (T) is treated, and receives treatment in the second period. The control unit (C) is never treated. 

---
# .center.pull[COVID-19 and Scented Candles]


&lt;img src="Slides_files/figure-html/unnamed-chunk-3-1.png" width="936" style="display: block; margin: auto;" /&gt;

---
# .center.pull[COVID-19 and Scented Candles]

&lt;img src="Slides_files/figure-html/unnamed-chunk-4-1.png" width="936" style="display: block; margin: auto;" /&gt;

---

# .center.pull[2x2 Difference-in-Differences]

- Building upon `\(\color{blue}{\text{Angrist &amp; Pischke (2008, p. 228)}}\)` we can think of these simple 2x2 DiDs as a fixed effects estimator.

- Potential Outcomes 
  - `\(Y_{i, t}^1\)` = value of dependent variable for unit `\(i\)` in period `\(t\)` with treatment.
  - `\(Y_{i, t}^0\)` = value of dependent variable for unit `\(i\)` in period `\(t\)` without treatment.
  
- The expected outcome is a *linear function* of unit and time fixed effects:
`$$E[{Y_{i, t}^0}] =\alpha_i + \alpha_t$$`
`$$E[{Y_{i, t}^1}] =\alpha_i + \alpha_t + \delta D_{st}$$`
- Goal of DiD is to get an unbiased estimate of the treatment effect `\(\delta\)`.

---
# .center.pull[Difference-in-Differences as Solving System of Equations for Unknown Variable]

- Difference in expectations for the *control* unit times t = 1 and t = 0:
`$$\begin{align*} E[Y_{C, 1}^0] &amp; = \alpha_1 + \alpha_C \\ E[Y_{C, 0}^0] &amp; = \alpha_0 + \alpha_C  \\ E[Y_{C, 1}^0] - E[Y_{C, 0}^0] &amp; = \alpha_1 - \alpha_0 \end{align*}$$`
 
- Now do the same thing for the *treated* unit:
   `$$\begin{align*} E[Y_{T, 1}^1] &amp; = \alpha_1 + \alpha_T + \delta \\ E[Y_{T, 0}^1] &amp; = \alpha_0 + \alpha_T  \\ E[Y_{T, 1}^1] - E[Y_{T, 0}^1] &amp; = \alpha_1 - \alpha_0 + \delta \end{align*}$$`
- If we assume the linear structure of DiD, then unbiased estimate of `\(\delta\)` is:

`$$\delta=
    \begin{align*} &amp; \left( E[Y_{T, 1}^1] - E[Y_{T, 0}^1] \right) - \left( E[Y_{C, 1}^0] - E[Y_{C, 0}^0] \right) \end{align*}$$`

---

# .center.pull[Two-Way Differencing]

&lt;img src="Slides_files/figure-html/unnamed-chunk-5-1.gif" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Regression DiD]
  
The DiD can be estimated through linear regression of the form:
  
`$$\tag{1} y_{it} = \alpha + \beta_1 TREAT_i + \beta_2 POST_t + \delta (TREAT_i \cdot POST_t) + \epsilon_{it}$$`
    
The coefficients from the regression estimate in (1) recover the same parameters as the double-differencing performed above:
`$$\begin{align*} 
\alpha &amp;= E[y_{it} | i = C, t = 0] = \alpha_0 + \alpha_C \\
\beta_1 &amp;= E[y_{it} | i = T, t = 0] - E[y_{it} | i = C, t= 0] \\ 
&amp;= (\alpha_0 + \alpha_T) - (\alpha_0 + \alpha_C) = \alpha_T - \alpha_C \\
\beta_2 &amp;= E[y_{it} | i = C, t = 1] - E[y_{it} | i = C, t = 0] \\ 
&amp;= (\alpha_1 + \alpha_C) - (\alpha_0 + \alpha_C) = \alpha_1 - \alpha_0 \\
\delta &amp;= \left(E[y_{it} | i = T, t = 1] - E[y_{it} | i = T, t = 0] \right) - \\
&amp;\hspace{.5cm} \left(E[y_{it} | i = C, t = 1] - E[y_{it} | i = C t = 0] \right) = \delta
\end{align*}$$`
    
---

# .center.pull[Regression DiD - The Workhorse Model]

- Advantage of regression DiD - it provides both estimates of `\(\delta\)` and standard errors for the estimates.

- `\(\color{blue}{\text{Angrist &amp; Pischke (2008)}}\)`:
  - "It's also easy to add additional (units) or periods to the regression setup... [and] it's easy to add additional covariates."

- Two-way fixed effects estimator:
`$$y_{it} = \alpha_i + \alpha_t + \delta^{DD} D_{it} + \epsilon_{it}$$`
  
  - `\(\alpha_i\)` and `\(\alpha_t\)` are unit and time fixed effects, `\(D_{it}\)` is the unit-time indicator for treatment.

  - `\(TREAT_i\)` and `\(POST_t\)` now subsumed by the fixed effects.

  - can be modified to include covariate matrix `\(X_{it}\)`, time trends, dynamic treatment effects estimation, etc. 
    
---
# .center.pull[COVID-19 and Scented Candles]


&lt;img src="Slides_files/figure-html/unnamed-chunk-6-1.png" width="936" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Regression DiD and Covid Candles]

- Let's apply the 2WFE DiD to the candle example:

--

- Our estimating model:

`$$R_{it} = \alpha_i + \alpha_t + \delta^{DD} D_{it} + \epsilon_{it}$$`

where:

- `\(R_{it}\)` is the rating within month `\(t\)` for candle type `\(i\)`. There are six types `\(i\)`, three scented and three unscented.

- `\(\alpha_i\)` are fixed effects for each type.

- `\(\alpha_t\)` are month-year fixed effects. 

- `\(D_{it}\)` is an indicator equal to 1 for the scented candle brands after Covid.

---
# .center.pull[Regression DiD and Covid Candles]

`\(\hspace{2cm}\)`

&lt;table class="table" style="margin-left: auto; margin-right: auto;"&gt;
 &lt;thead&gt;
&lt;tr&gt;&lt;th style="border-bottom:hidden; padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"&gt;&lt;div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; "&gt;TWFE DiD Estimate&lt;/div&gt;&lt;/th&gt;&lt;/tr&gt;
  &lt;tr&gt;
   &lt;th style="text-align:center;"&gt;   &lt;/th&gt;
   &lt;th style="text-align:center;"&gt;   &lt;/th&gt;
   &lt;th style="text-align:center;"&gt;   &lt;/th&gt;
  &lt;/tr&gt;
 &lt;/thead&gt;
&lt;tbody&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt; Post-Covid * Scented Candle &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; \(\hat{\delta}\) &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; -0.228 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt;  &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; \(\sigma \) &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 0.057 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt;  &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; t &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; -3.971 &lt;/td&gt;
  &lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;

---
# .center.pull[Where It Goes Wrong]

- Developed literature now on the issues with TWFE DiD with "staggered treatment timing" &lt;span style="color:blue"&gt; (Abraham and Sun (2018), Borusyak and Jaravel (2018), Callaway and Sant'Anna (2019), Goodman-Bacon (2019), Strezhnev (2018), Athey and Imbens (2018))&lt;span&gt;

  - Different units receive treatment at different periods in time.

--

- Probably the most common use of DiD today. If done right can increase amount of cross-sectional variation.

--
- Without digging too far into the literature: 
  
  - `\(\delta^{DD}\)` with staggered treatment timing is a *weighted average of many different treatment effects*. 
  
  - We know little about how it measures when treatment timing varies, how it compares means across groups, or why different specifications change estimates.
  
  - The weights are often negative and non-intuitive.
  
---

# .center.pull[Staggered DiD in Accounting]

&lt;table class="table table-striped" style="margin-left: auto; margin-right: auto;"&gt;
 &lt;thead&gt;
  &lt;tr&gt;
   &lt;th style="text-align:center;"&gt; Journal &lt;/th&gt;
   &lt;th style="text-align:center;"&gt; DiD &lt;/th&gt;
   &lt;th style="text-align:center;"&gt; Staggered &lt;/th&gt;
   &lt;th style="text-align:center;"&gt; Staggered Percentage &lt;/th&gt;
  &lt;/tr&gt;
 &lt;/thead&gt;
&lt;tbody&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt; JAR &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 52 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 21 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 40.38% &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt; JAE &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 63 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 34 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 53.97% &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt; TAR &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 108 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 52 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 48.15% &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt; RAST &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 46 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 24 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 52.17% &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center;"&gt; CAR &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 43 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 17 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 39.53% &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr grouplength="2"&gt;&lt;td colspan="4" style="border-bottom: 1px solid;"&gt;&lt;strong&gt;Combined&lt;/strong&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;
   &lt;td style="text-align:center; padding-left: 2em;" indentlevel="1"&gt; Top 3 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 223 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 107 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 47.98% &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:center; padding-left: 2em;" indentlevel="1"&gt; Total &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 312 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 148 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 47.44% &lt;/td&gt;
  &lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;


---

# .center.pull[Bias with TWFE - Goodman-Bacon (2019)]

- Important Insights

  - `\(\delta^{DD}\)` is just the weighted average of all the 2x2 sub-group comparison treatment effects. The weights are a function of the size of the subsample, relative size of treatment and control units, and the timing of treatment in the sub sample.

  - **Already-treated units act as controls even though they are treated.**

  - Given the weighting function, panel length alone can change the DiD estimates substantially, even when each `\(\delta^{DD}\)` does not change.

  - Groups treated closer to middle of panel receive higher weights than those treated earlier or later.

---

# .center.pull[Bias with TWFE - Goodman-Bacon (2019)]

Main Decomposition:

`$$\underset{N \rightarrow \infty}{\text{plim }} \hat{\delta}^{DD} = \color{green}{VWATT} + \color{blue}{VWCT} -\color{red}{\Delta ATT}$$`

--
`\(\hspace{2cm}\)`

- `\(\color{green}{VWATT}\)` = variance-weighted average treatment effect on the treated.

--

`\(\hspace{2cm}\)`
- `\(\color{blue}{VWCT}\)` = variance-weighted common trend. 

--

`\(\hspace{2cm}\)`
- `\(\color{red}{\Delta ATT}\)` = weighted sum of the *change* in treatment effects within each unit's post-period with respect to another unit's treatment timing.

---
# .center.pull[Simulation Exercise]

- Can show how easily `\(\delta^{DD}\)` goes awry up through a simulation exercise.

--

- Assume we're modeling outcome variable `\(y_{it}\)` on balanced panel with `\(T = 36\)` years from 1980 to 2015 with 1000 firms `\(i\)`.

--

- Time-invariant unit effects and time-varying year effects drawn from `\(\sim N \left(0, \frac{1}{2}^2\right)\)`.

--

- Firms are incorporated in of 50 randomly drawn states, and states are randomly assigned into three treatment groups `\(G_g \in \{1989, 1998, 2006\}\)`.

---

# .center.pull[Simulation Exercise]

`\(\hspace{0.5cm}\)`

**Model the treatment effect process in three ways:**

--

1) Only one treatment period (1998) and one treated group, with constant additive treatment effects.
  
--

2) Allow for staggered treatment timing but with constant additive effects. Simulated treatment effects `\(\tau\)` are all positive in expectation but decrease over time  `\(\left( \tau_{G1989} = 5, \tau_{G1998} = 3, \tau_{G2007} = 1 \right)\)`.

--

3) Allow for both staggered treatment timing and change-in-trend "dynamic" treatment effects. Instead of a constant `\(\tau\)` for each group, `\(\tau_i\)` is the yearly increase in outcome variable that compounds over time. Here `\(\tau_{i, G1989} = 0.5, \tau_{i, G1998} = 0.3,\)` and `\(\tau_{i, G2007} = 0.1\)`.

---
# .center.pull[Simulation Exercise]

&lt;img src="Slides_files/figure-html/unnamed-chunk-9-1.png" width="1008" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Simulation Exercise]

`\(\hspace{0.5cm}\)`

- With the simulated data we estimate TWFE DiD using MLE on:

`$$y_{it} = \alpha_i + \alpha_t + \delta^{DD}D_{it} + \epsilon_{it}$$`

`\(\hspace{0.5cm}\)`

- Simple regression model with unit and time fixed effects.

`\(\hspace{0.5cm}\)`

- For each of the three simulated datasets we run a Monte Carlo simulation where we create the datasets 1,000 times and plot the distribution of `\(\widehat{\delta^{DD}}\)`.

`\(\hspace{0.5cm}\)`

- Bias is deviation from true underlying treatment effect.

---
# .center.pull[Simulation Exercise]

&lt;img src="Slides_files/figure-html/unnamed-chunk-10-1.png" width="864" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Goodman-Bacon Decomposition for Simulation 3]

`\(\hspace{0.5cm}\)`


&lt;iframe src="p.html" width="800" height="500" scrolling="yes" seamless="seamless" frameBorder="0"&gt; &lt;/iframe&gt;

---

# .center.pull[Remedies]

---

# .center.pull[Callaway &amp; Sant'Anna]

- Inverse propensity weighted long-difference in cohort-specific average treatment effects between treated and untreated units for a given treatment cohort. 

`$$\begin{equation} ATT(g, t) = \mathbb{E} \left[\left( \frac{G_g}{\mathbb{E}[G_g]} - \frac{\frac{p_g(X)C}{1 - p_g(X)}}{\mathbb{E}\left[\frac{p_g(X)C}{1 - p_g(X)} \right]} \right) \left(Y_t - T_{g - 1}\right)\right] \end{equation}$$`

- Without covariates, as in the simulated example here, it calculates the simple long difference between all treated units `\(i\)` in relative year `\(k\)` with all potential control units that have not yet been treated by year `\(k\)`.

---
# .center.pull[Abraham and Sun]
  
- A relatively straightforward extension of the standard event-study TWFE model:
  
  `$$y_{it} = \alpha_i + \alpha_t + \sum_e \sum_{l \neq -1} \delta_{el}(1\{E_i = e\} \cdot D_{it}^l) + \epsilon_{it}$$`
  
- You saturate the relative time indicators (i.e. t = -2, -1, ...) with indicators for the treatment initiation year group, and aggregate to overall relative time indicators weighting by cohort size.

- In the case of no covariates, this gives you the same estimate as Callaway &amp; Sant'Anna if you *fully saturate* the model with time indicators (leaving only two relative year identifiers missing).

- The authors don't claim that it can be used with covariates, but it seemingly follows if we think it is okay with normal TWFE DiD. 

---
# .center.pull[Stacked Regression]
  
- Similar to the standard TWFE DiD, but we ensure that no previously treated units enter as controls by trimming the sample.

- For each treatment cohort `\(G_g\)`, get all treated units, and all units that are not treated by year `\(g + k\)` where `\(g\)` is the treatment year and `\(k\)` is the outer most relative year that you want to test (e.g. if you do an event study plot from -5 to 5, `\(k\)` would equal 5).

- Keep only observations within years `\(g - k\)` and `\(g + k\)` for each cohort-specific dataset, and then stack them in relative time. 

- Run the same TWFE estimates as in standard DiD, but include interactions for the cohort-specific dataset with all of the fixed effects, controls, and clusters.

---
# .center.pull[Simulations - Remedies]

&lt;img src="Slides_files/figure-html/unnamed-chunk-12-1.png" width="864" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Replication]

---
# .center.pull[Big Bad Banks]

- Beck, Thorsten, Ross Levine, and Alexey Levkov, 2010, "Big Bad Banks? The Winners and Losers from Bank Deregulation in the United States", *Journal of Finance* 65 (5).

**Story:**

- 1970s through the 1990s, most states removed restrictions on intrastate branching.

--

- Intensified bank competition and improved bank performance.

--

- Theoretically unclear what the distributional effects of bank deregulation would be.

--

- "Exploiting the cross‐state, cross‐time variation in the timing of branch deregulation, we find that deregulation materially tightened the distribution of income."

---
# .center.pull[Deregulatory Timing]

&lt;img src="Slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" /&gt;


---
# .center.pull[Baseline Model]

`$$\text{Log(Gini)} = \alpha_i + \alpha_t + \delta D_{it} + \epsilon_{it}$$`

where Log(Gini) is the natural log of the state level Gini coefficient from CPS, `\(\alpha_i\)` and `\(\alpha_t\)` are state and year fixed effects, and `\(D_{it}\)` is an indicator variable for years after deregulation.

`\(\hspace{2cm}\)`

&lt;table class="table" style="margin-left: auto; margin-right: auto;"&gt;
&lt;caption&gt;The Impact of Deregulation on Income Inequality&lt;/caption&gt;
 &lt;thead&gt;
&lt;tr&gt;&lt;th style="border-bottom:hidden; padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"&gt;&lt;div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; "&gt;Panel A: No Controls&lt;/div&gt;&lt;/th&gt;&lt;/tr&gt;
  &lt;tr&gt;
   &lt;th style="text-align:left;"&gt;   &lt;/th&gt;
   &lt;th style="text-align:center;"&gt; Log Gini &lt;/th&gt;
  &lt;/tr&gt;
 &lt;/thead&gt;
&lt;tbody&gt;
  &lt;tr&gt;
   &lt;td style="text-align:left;"&gt; Bank deregulation &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; -0.022 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:left;"&gt;  &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; (0.007)*** &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:left;"&gt; Adjusted R-Squared &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 0.51 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:left;"&gt; Observations &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 1519 &lt;/td&gt;
  &lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;

---
# .center.pull[Goodman-Bacon Decomposition]

`\(\hspace{2cm}\)`

&lt;table class="table" style="margin-left: auto; margin-right: auto;"&gt;
&lt;caption&gt;Goodman-Bacon Decomposition of Table II Results&lt;/caption&gt;
 &lt;thead&gt;
  &lt;tr&gt;
   &lt;th style="text-align:left;"&gt; Type &lt;/th&gt;
   &lt;th style="text-align:center;"&gt; Weighted 
 Average &lt;/th&gt;
   &lt;th style="text-align:center;"&gt; Total 
 Weight &lt;/th&gt;
  &lt;/tr&gt;
 &lt;/thead&gt;
&lt;tbody&gt;
  &lt;tr&gt;
   &lt;td style="text-align:left;"&gt; Earlier vs Later Treated &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 0.005 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 0.143 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style="text-align:left;"&gt; Later vs Earlier Treated &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; -0.027 &lt;/td&gt;
   &lt;td style="text-align:center;"&gt; 0.857 &lt;/td&gt;
  &lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;

---
# .center.pull[Goodman-Bacon Decomposition]

&lt;img src="Slides_files/figure-html/unnamed-chunk-16-1.png" width="864" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Event Study Specification]

`$$\text{log(Gini)}_{it} = \alpha_i + \alpha_t + \beta_1 D_{it}^{-\overline{10}} + \beta_2 D_{it}^{-9} \ldots \beta_{25}D_{it}^{+\overline{15}} + \epsilon_{st}$$`
&lt;img src="Slides_files/figure-html/unnamed-chunk-17-1.png" width="720" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Modified Event Study Specification]

&lt;img src="Slides_files/figure-html/unnamed-chunk-18-1.png" width="720" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Remedies]

&lt;img src="Slides_files/figure-html/unnamed-chunk-19-1.png" width="864" style="display: block; margin: auto;" /&gt;

---
# .center.pull[Conclusion]

`\(\hspace{2cm}\)`

- A well-known and very commonly used shock that has reasonable treatment timing dispersion. 

`\(\hspace{2cm}\)`
--

- Decomposition from Goodman-Bacon shows the result is driven by problematic comparisons.

`\(\hspace{2cm}\)`
--

- Once correcting for staggering and treatment effect heterogeneity - no effect.
    </textarea>
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